#### Abstract

Let , let , and let be an analytic self-map of and . The boundedness and compactness of generalized composition operators , from () spaces to spaces are investigated.

#### 1. Introduction and Preliminaries

Let be an analytic self-map of the open unit disc of the complex plane . Let be the space of all analytic functions in and . If is a Banach space, then we denote the unit ball in by . For , .

A positive continuous function on the interval is called normal if there exist three constants and such that

A function belongs to the Bloch type space if where is normal and radial and . The space is a Banach space with the norm .

The little Bloch type space consists of all such that For , , is the -Bloch space ; for , is the classical Bloch space; for example, see [1].

For , , , is a nondecreasing function, and is a given reasonable function. An analytic function on is said to belong to in [2] if and an analytic function if where denotes the normalized Lebesgue area measure on , is a green function, and .

classes are more general than many classes of analytic functions and have attracted a lot of attention in recent years. When , . When , , , and . When , and . Moreover, the following results hold:(1);(2) if and only if where

The composition operator is defined by , . This operator has been studied for many years. The first setting was in the Hardy space , the space of functions analytic on (see [3]). Madigan and Matheson (see [1]) gave a characterization of the compact composition operators on the Bloch space . For more details, see [4–12]. In [13], Li and Stević defined the generalized composition operator as follows: The operator induces many known operators. When , the operator is essentially (up to a constant) the composition operator . When , the operator coincides with the operator defined by So the generalized composition operator can be considered as a generalization of the composition operator and the operator .

A fundamental problem in the study of generalized composition operators is to investigate the relations between function theoretic properties of and and operator theoretic properties of the restriction of to various Banach spaces of analytic functions. A lot of attentions have been attracted to study the problem on many Banach spaces of analytic functions in recent years. In [9], the authors studied composition operators from Bloch type spaces into spaces. In [14], the authors characterized the boundedness and compactness of generalized composition operators on spaces. In [15], Rezaei and Mahyar studied generalized composition operators from logarithmic Bloch type spaces to type spaces. In [16], essential norms of generalized composition operators from Bloch type spaces to type spaces were given. In [17], generalized composition operators from spaces to Bloch-type spaces were characterized. In [18], Stević investigated generalized composition operators between mixed-norm space and some weighted spaces and from logarithmic Bloch spaces to mixed-norm spaces. In [3], Zhang and Liu studied generalized composition operators from Bloch type spaces to type spaces. In [19], generalized composition operator acting from Bloch-type spaces to mixed-norm space was studied. In [12], generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces were investigated. In [20], generalized composition operators and Volterra composition operators on Bloch spaces on the unit ball were studied. This paper is devoted to investigating the boundedness and compactness of generalized composition operators from () spaces to spaces. Throughout this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other.

#### 2. Main Results and Their Proofs

To derive our results, we need the following lemmas.

Lemma 1. *Assume that , , is a nonnegative nondecreasing function on , and is a given reasonable function. Assume that is a normal function, is an analytic self-map of , and . Then is compact if and only if, for every bounded sequence in which converges to uniformly on compact subsets of , .*

Lemma 1 can be proved in a standard way of Theorem 3.11 in [4].

The following lemma is similar to Lemma 2.2 in [5, 7], using the results for the Hadamard gap series and following a technique used before in the Bloch space in [5, 7]. Specific details can be seen in [9].

Lemma 2. *Let be a nonincreasing radial weight function and normal on the interval . Then there exist two functions such that, for each ,
*

Theorem 3. *Assume that , , is an analytic self-map of , is a normal function, is nonnegative and nondecreasing in , and is a given reasonable function. Then the following statements are equivalent:*(a)* is bounded;*(b)* is bounded;*(c)

*Proof. *(a) (b) Since , then (a) implies (b).

(b) (c) Suppose (b) holds; then for all . For any given , the function , , belongs to and . Let , be the functions from Lemma 2 and we get
Then (11) holds with Fatou’s Lemma.

(c) (a) For ,

Theorem 4. *Assume that , , is an analytic self-map of , is a normal function, is nonnegative and nondecreasing in , and is a given reasonable function. Then the following statements are equivalent:*(a)* is compact;*(b)* is compact;*(c)

*Proof. *(a) (b) Since , then (a) implies (b).

(b) (c) Assume that (b) holds; then we have (14), Let
Then is bounded in and uniformly on the compact subsets of as . Since is compact, then by Lemma 1
This means, for any given , there exists such that implies
Hence, for ,
Choosing such that , then

For , let for . Then and uniformly on compact subsets of as . Since is compact, then as . Then for every there exists such that
By the triangle inequality, then
which means, for any and , there exists such that for

Since is compact, is relatively compact in ; then there are finite functions such that, for any and , we can find satisfying

Take . Then for

Then

Hence, we have shown that for any there exists such that for all

Let , be the functions in Lemma 2; then, for , the functions are included in . Thus by Lemma 2 and Fatou’s Lemma, we get (15).

(c) (a) Assume that (14) and (15) hold. Assume that is a bounded sequence in such that uniformly on compact subsets of . Assume ; by (15), for any given , there exists such that

Since uniformly on compact subsets of , then uniformly on compact subsets of . Then for above , there exists such that implies for . Thus,

Hence, as . Thus is compact.

*Remark 5. *For , , is the -Bloch space . Let and in Theorems 3 and 4; we easily obtain the following results in [3].

Corollary 6. *Assume that , , , is an analytic self-map of , and is a nonnegative nondecreasing function on . Then the following statements are equivalent:*(a)* is bounded;*(b)* is bounded;*(c)

Corollary 7. *Assume that , , , is an analytic self-map of , and is a nonnegative nondecreasing function on . Then the following statements are equivalent:*(a)* is compact;*(b)* is compact;*(c)

*Remark 8. *As , the operator is essentially the composition operator , since the difference is constant. Moreover, ; . Let and in Theorems 3 and 4; we easily obtain the following results in [9].

Corollary 9. *Assume that , , is an analytic self-map of , is a normal function, and is nonnegative and nondecreasing in . Then the following statements are equivalent:*(a)* is bounded;*(b)* is bounded;*(c)

Corollary 10. *Assume that , , is an analytic self-map of , is a normal function, and is nonnegative and nondecreasing in . Then the following statements are equivalent:*(a)* is compact;*(b)* is compact;*(c)* and
*

*Problem 11. *Can the boundedness and compactness of the generalized composition operator be characterized by use of function theoretic properties of and ?

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to express their sincere thanks to the referees for careful reading and suggestions which helped them improve the paper. This work was supported in part by the National Natural Science Foundation of China (nos. 11201127 and 11271112), the Young Core Teachers Program of Henan Province (no. 2011GGJS-062), and the Natural Science Foundation of Henan Province (nos. 122300410110 and 2010A110009).